On Markuševič decompositions of Banach spaces (Q1174864)

A sequence \((G_ n)\) of closed linear subspaces of a Banach space \(E\) is said to be a generalized decomposition of \(E\) if there is a sequence \((v_ n)\) of linear projections on \(E\) with \(v_ n(E)=G_ n\) for each \(n\), such that \(v_ iv_ j=0\) whenever \(i\neq j\) and \((v_ n)\) is total over \(E\). A generalized decomposition \((G_ n)\) is said to be a Markuševič decomposition (\(M\)-decomposition) of \(E\) if the linear span of \(\bigcup G_ n\) is dense in \(E\). Further, the sequence \((G_ n)\) is said to be a Schauder decomposition of \(E\) if for each \(x\in E\) there exists an unique sequence \((x_ n)\) with \(x_ n\in G_ n\) such that \(x=\sum_{n=1}^ \infty x_ n\). The objective of the paper is to study \(M\)-decompositions. The associated sequence of projections \((v_ n)\) to the generalized decomposition is unique if and only if \((G_ n)\) is an \(M\)-decomposition. The existence of \(M\)-decompositions in several Banach spaces, for instance the weakly compactly generated spaces has been established. A characterization of \(M\)-decompositions in terms of its subsequences and that of Schauder decompositions in terms of \(G\)-decompositions have been obtained. Finally, the twin relationship between the \(M\)-decompositions of a Banach space with certain corresponding decompositions in the conjugate space has been proved.